Cryptographic Method and System for Secure Authentication and Key Exchange

1. A computer-implemented method of authentication and establishment of a communication channel between client devices over a network, comprising: a) receiving, from a first client device, a randomly generated binary string; b) combining, by a second client device, a secret key associated with the first client device and the second client device with the randomly generated binary string to form a new binary string; c) initiating, by the second client device, a binary pair (A, B) with an initial value of (A 0 , B 0 ); d) reading, by the second client device, a bit of the new binary string; e) adding, by the second client device, a current value of B to A when the read bit is zero; f) adding, by the second client device, a current value of A to B when the read bit is one; g) reading, by the second client device, a next bit of the new binary string; repeating, via processing circuitry, steps e) through g) until all bits of the new binary string have been read; computing, by the second client device, a final binary pair (A, B) after all of the bits have been read; generating, by the second client device, a first ephemeral Diffie-Hellman (DH) public key as a function of the final binary pair (A, B); receiving, at the second client device from the first client device, a second ephemeral DH public key; decrypting, at the second client device, the second ephemeral DH public key as a function of the final binary pair (A, B); exchanging, between the first client device and the second client device, challenge responses based on decryption of the first and second ephemeral DH public keys; authenticating, between the first client device and the second client device, the challenge responses; agreeing, based on authentication of the challenge responses, on a session key for communication between the first client device and the second client device, wherein A, is a current value of A and is equal to A i−1 when the read bit is equal to one; and establishing a secure communication channel over the network, between the first client device and the second client device, wherein information is communicated between the first client device and the second client device over the secure communication channel based on the agreed session key.

2. The computer-implemented method of claim 1 , wherein the final binary pair (A, B) contains two unique prime integers.

3. The computer-implemented method of claim 1 , wherein the final binary pair (A, B) has a modular multiplicative inverse.

4. The computer-implemented method of claim 1 , wherein the binary string is variable.

5. The computer-implemented method of claim 4 , wherein the final binary pair (A, B) varies upon adding, changing, or deleting one or more bits of the binary string.

6. The computer-implemented method of claim 1 , wherein prior knowledge of the secret key by the first client device and the second client device is required for authentication.

7. The computer-implemented method of claim 1 , wherein the computer-implemented method reveals no information about the secret key to other client devices.

8. The computer-implemented method of claim 7 , wherein the computer-implemented method reveals no information about the session key to the other client devices.

9. A non-transitory computer-readable medium having computer-readable instructions thereon, that when executed by a computer, cause the computer to perform a method of authentication and establishment of a communication channel between client devices over a network, the method comprising: a) receiving, from a first client device, a randomly generated binary string; b) combining, by a second client device, a secret key associated with the first client device and the second client device with the randomly generated binary string to form a new binary string; c) initiating, by the second client device, a binary pair (A, B) with an initial value of (A 0 , B 0 ); d) reading, by the second client device, a bit of the new binary string; e) adding, by the second client device, a current value of B to A when the read bit is zero; f) adding, by the second client device, a current value of A to B when the read bit is one; g) reading, by the second client device, a next bit of the new binary string; repeating, via processing circuitry, steps e) through g) until all bits of the new binary string have been read; computing, by the second client device, a final binary pair (A, B) after all of the bits have been read; generating, by the second client device, a first ephemeral Diffie-Hellman (DH) public key as a function of the final binary pair (A, B); receiving, at the second client device from the first client device, a second ephemeral DH public key; decrypting, at the second client device, the second ephemeral DH public key as a function of the final binary pair (A, B); exchanging, between the first client device and the second client device, challenge responses based on decryption of the first and second ephemeral DH public keys; authenticating, between the first client device and the second client device, the challenge responses; agreeing, based on authentication of the challenge responses, on a session key for communication between the first client device and the second client device, wherein A i is a current value of A and is equal to A i−1 when the read bit is equal to one; and establishing a secure communication channel over the network, between the first client device and the second client device, wherein information is communicated between the first client device and the second client device over the secure communication channel based on the agreed session key.

10. The non-transitory computer-readable medium of claim 9 , wherein the method is based on an addition operation applied to the binary string.

11. The non-transitory computer-readable medium of claim 9 , wherein the method receives the binary string as input and generates two unique prime integers as output.

12. The non-transitory computer-readable medium of claim 9 , wherein the binary string is based on an input string.

13. The non-transitory computer-readable medium of claim 9 , wherein B i is a current value of B and is equal to B i−1 +A i−1 when the read bit is equal to one.

14. The non-transitory computer-readable medium of claim 13 , wherein A i is equal to A i−1 +B i−1 when the read bit is equal to zero.

15. The non-transitory computer-readable medium of claim 14 , wherein B i is equal to B i−1 when the read bit is equal to zero.